Idea

The little kk-disk operad or little kk-cubes operad (to distinguish from the framed little n-disk operad) is the topological operad/(∞,1)-operadEkE_k whose nn-ary operations are parameterized by rectilinear disjoint embeddings of nnkk-dimensional cubes into another kk-dimensional cube.

When regarded as a topological operad, the topology on the space of all such embedding is such that a continuous path is given by continuously moving the images of these little cubes in the big cube around.

More generally, if SS is a ﬁnite set, then we will say that a map □k×S→□k\square^k
\times S \to \square^k is a rectilinear embedding if it is an open embedding whose restriction to each connected component of □k×S\square^k\times S is rectilinear.

Let Rect(□k×S,□k)Rect(\square^k \times S, \square^k ) denote the collection of all rectitlinear embeddings from □k×S\square^k \times S into □k\square^k . We will regard Rect(□2×S,□k)Rect(\square^2\times S, \square^k ) as a topological space (it can be identiﬁed with an open subset of (R2k)S)(\mathbf{R}^{2k} )^S ).

The spaces Rect(□k×{1,...,n},□k)Rect(\square^k \times \{1, . . . , n\}, \square^k) constitute the nn-ary operations of a topological operad, which we will denote by tEktE_k and refer to as the little k-cubes operad.

Say an 𝔼[1]\mathbb{E}[1]-algebra object is grouplike if it is grouplike as an AssAss-monoid. Say that an 𝔼[k]\mathbb{E}[k]-algebra object in 𝒳\mathcal{X} is grouplike if the restriction along 𝔼[1]↪𝔼[k]\mathbb{E}[1] \hookrightarrow \mathbb{E}[k] is. Write

Proof

Specifically for 𝒳=Top\mathcal{X} = Top, this refines to the classical theorem by (May).

Theorem (May recognition theorem)

Let YY be a topological space equipped with an action of the little cubes operad𝒞k\mathcal{C}_k and suppose that YY is grouplike. Then YY is homotopy equivalent to a kk-fold loop space ΩkX\Omega^k X for some pointed topological space XX.